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The polynomial f(x) satifies the equation f(x)-f(x-2)=5x+1 for all x. If p and q are the coefficients of x^2 and x respectively, in f (x), then what is the value of p+q?

 Jun 13, 2021
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Since $f(x)-f(x-2)=5x+1$, a linear function, $f$ must be a quadratic function.  Let $f$ be defined by $f(x)=px^2+qx+r$ for some constants $p$, $q$, $r$.  By substituting $x$ and $x-2$ into the formula for $f(x)$ we get $f(x)-f(x-2)=4px-4p+2q=5x+1$ for all $x$.  This gives two equations $4p=5$ and $-4p+2q=1$.  Solve for $p$ and $q$ and substitute them into $p+q$.
 

 Jun 13, 2021

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